Stratified Morse Theory with Tangential Conditions
arXiv:math/0310019
Abstract
Our objective is to develop a stratified Morse theory with tangential conditions. We define a continuous strata-wise smooth Morse function on an abstract stratified space by using control conditions and radiality assumptions on the gradient vector field. For critical points of a Morse function one can show that the local unstable set is a manifold and the local stable set is itself an abstract stratified space. We also give a normal form for the gradient dynamics in the neighborhood of critical points. For a stratified Morse pair which satisfies the generic Morse-Smale condition we can build the Morse-Witten complex and show that its homology is equivalent to the singular homology of the space.
This paper has been withdrawn by the author. Part of this article has been published in revised form under the title "Morse-Smale-Witten complex for gradient-like vector fields on stratified spaces" in Chéniot, Denis (ed.) et al., Singularity theory. Proceedings of the 2005 Marseille singularity school and conference